I found

\(\displaystyle P(2ct,0)\) , \(\displaystyle Q(0,\frac{2c}{t})\)

and \(\displaystyle R(\frac{c(t^2+1)}{t},\frac{c(t^2+1)}{t})\)

\(\displaystyle S(\frac{c(t^2-1)}{t},-\frac{c(t^2-1)}{t})\)

I tried the gradients of the diagonals PQ and RS are perpendicular, and got no restrictions on \(\displaystyle t^2\) I also tried finding the lengths of all the sides but they are all the same \(\displaystyle \frac{c}{t}\sqrt{t^4+1}\) so i cannot prove that PRQS is a rhombus unless \(\displaystyle t^2=1\).

Thanks